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Rückert, H. (2007, June 26). Dialogues as a Dynamic Framework for Logic. Retrieved from https://hdl.handle.net/1887/12099
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Dialogues as a
Dynamic Framework for Logic
Helge Rückert
Dialogues as a
Dynamic Framework for Logic
Proefschrift ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. mr. P.F. van der Heijden, volgens besluit van het College voor Promoties
te verdedigen op dinsdag 26 juni 2007 klokke 10 uur
door Helge Rückert
geboren te Illingen (Duitsland) in 1971
Prof. dr. J.P. Van Bendegem (Vrije Universiteit Brussel) Prof. dr. H. Wansing (Technische Universität Dresden) Dr. J.B.M. van Rijen
The research reported herein was financially supported by the University of Saarbrücken and the DAAD (Deutscher Akademischer Austauschdienst).
Rückert, H.
‘Why Dialogical Logic?’ (English) 15
Rahman, S., Rückert, H. and Fischmann, M.
‘On Dialogues and Ontology. The Dialogical Approach to Free Logic’ (English) 31
Rahman, S. and Rückert, H.
‘Dialogische Modallogik (für T, B, S4 und S5)’ (German) 45
Rahman, S. and Rückert, H.
‘Dialogische Logik und Relevanz’ (German) 79
Rahman, S. and Rückert, H.
‘Dialogical Connexive Logic’ (English) 105
Rahman, S. and Rückert, H.
‘Eine neue dialogische Semantik für lineare Logik’ (German) 141
Rückert, H.
‘Logiques Dialogiques ‘Multivalentes’’ (French) 173
Bibliography 199
Summary/Samenvatting (English/Dutch) 207
CV (English) 209
answer says that the meaning of an expression is some sort of entity associated with the expression. In the field of logic this idea leads to all kinds of model-theoretic semantics.
Some, unhappy with such an account of meaning because of its realistic flavour and ontological commitments, prefer an alternative account of meaning that relates meaning to use.1 Applied to logic, this idea leads to two different kinds of approaches. According to the first the meaning of the logical particles is given via rules that determine how these particles can be used in proofs (proof-theoretical semantics). The second one characterises the meaning of the logical particles via rules that determine how these particles can be used in certain games, argumentation games (game-theoretical semantics). Within the game-theoretical approach three main traditions can be distinguished:2
(1) The constructivist approach of Paul Lorenzen and Kuno Lorenz, who sought to overcome the limitations of Operative Logic by providing dialogical foundations to it. The method of semantic tableaux for classical and intuitionistic logic as introduced by Evert W. Beth (1955) could thus be identified as a method for the notation of winning strategies of particular dialogue games (cf. Lorenzen/Lorenz (1978), Lorenz (1981), Felscher (1986)).
(2) The game-theoretical approach of Jaakko Hintikka who recognised at a very early stage that a two-players semantics offers a new dynamic device for studying logical systems. This approach is better known and opened many new research lines developed by Hintikka and co-authors, specially the semantic games which offer a deep and thorough insight into the notion of scope implemented by the Independence-Friendly Logic, the interrogative games which are essentially epistemic games and the formal games of theorem-proving which deal with logical truth of propositions and not with their material truth (as in the semantic games) or with one’s knowledge of their truth (as in the epistemic games) Cf.
Hintikka/Sandu (1996) and Hintikka (1996, 1996-1998).
(3) The argumentation theory approach of Else Barth and Erik Krabbe (1982) (cf. also Gethmann (1979)), who sought to link dialogical logic with the informal logic or Critical Reasoning originated by the seminal work of Chaim Perelman (cf.
Perelman/Olbrechts-Tyteca (1958)), Stephen Toulmin (1958), Arne Naess (1966)
1 A main source of inspiration for such a view is the later Wittgenstein with his language games. His most explicit formulation of the idea of meaning as use, is the following: “For a large class of cases – though not for all – in which we employ the word “meaning” it can be defined thus: the meaning of a word is its use in language.”
(Wittgenstein (1953, §43))
2 The following three paragraphs are taken from Rahman/Rückert (2001c).
and Charles Hamblin (1970) and developed further by Ralph Johnson (1999), Douglas Walton (1984), John Woods (1988) and associates.
Unfortunately, for quite a long time, these three traditions followed separate paths and with some occasional exceptions did not actually pool their results in a common project.3
There are also interesting connections between the dialogical approach to logic and two other influential, more recent research programs:
1) Game Theory and Logic
One characteristic feature of the dialogical approach is that logic is studied via certain games (for example in the case of classical or intuitionistic logic via two- person zero-sum games with perfect information). So, the main interest is logical systems, and the main tool for studying them are certain games. On the other hand there is a whole mathematical discipline devoted to the systematic and formal study of games themselves: game theory. In game theory the games are the objects of investigation.
Given the above connection between logical systems and games which is characteristic for dialogical logic and other related approaches, it is a promising idea to look systematically at results from game theory and examine whether they lead, via the bridge from games to logic, to important logical results, too. This idea constitutes the basis for the research done by a whole group of logicians and other scientists under the guidance of Johan van Benthem at the ILLC (Institute for Logic, Language and Computation) in Amsterdam.4
2) Adaptive Logic
One big advantage of dialogical logic is that it easily allows for the combination of different logical systems. In general this is possible because very often different specific structural rules are decisive in order to obtain different logical systems.
So, if one uses both (or more) of these specific structural rules this results in a new logical system which is in some sense a combination of these two (or more) logics.
There exists another very interesting framework for the systematic combination of different logical systems: so-called adaptive logic which was developed by Diderik Batens. This framework makes it possible to integrate two logics (one is called the upper limit logic, the other one the lower limit logic) into one proof system with a well defined consequence relation.5
I think, it might be very rewarding to compare dialogical logic and adaptive
3 Quite recently there has been a lot of interaction between the first two, however. Witness Rahman/Tulenheimo (2007), Rahman (2007) and Degremont/Rahman (2007)
4 An enormous number of publications resulted from this big research program. Just to mention one: Van Benthem (2007)
5 Again there are lots of publications by Diderik Batens and his collaborators. So, let me just give the URL of the Adaptive Logics Home Page: http://logica.ugent.be/adlog/al.html.
logic with respect to the similarities and the differences in the ways how logical systems are combined in these two general logical frameworks. A first step in this direction has already been done by Rahman/Van Bendegem (2002).
This is a collection of seven papers on dialogical logic, most of them co-authored by Shahid Rahman. It starts with the overview paper ‘Why Dialogical Logic?’ which could serve as an introduction to the other six papers. Those other papers, ordered according to the time when they had been written (which is different from the order of their first publication) are related by one central theme. As each of them presents dialogical formulations of a different non- classical logic, they show that dialogical logic provides a powerful and flexible general framework for the development and study of various logical formalisms and combinations thereof. So, especially for logical pluralists who do not believe in “the one and only correct logic”, but who take an instrumentalist attitude towards logic and see different logical systems as formal tools for different purposes, dialogical logic is an interesting and fruitful alternative to model-theoretic and other logical methodologies.
Most of the papers were first published in an internal memo series at the University of Saarbrücken before they were submitted to an international journal. For this volume all the papers have been corrected, clarified and amended at various places. Furthermore the three German papers have been adapted to new German orthography and stylistically revised. The layout and composition of the papers has been unified and a single bibliography for the whole volume can now be found at the end. As each paper is supposed to be self-contained and can be read independently from the others, it was impossible to avoid some redundancies, especially when presenting the basic features of standard dialogical logic in each paper. Here are some remarks on each of the papers assembled in this collection:
Rückert, H.: ‘Why Dialogical Logic?’, in: Wansing, H. (ed.): Essays on Non-Classical Logic (Advances in Logic - Vol. 1), New Jersey, London, Singapore, Hong Kong 2001, p.
165-185
This paper grew out of a German talk called “Wodurch sich der dialogische Ansatz in der Logik auszeichnet” I had given on occasion of the XVIII. German Congress for Philosophy, Constance, October 1999. An extended abstract of this talk was published in the congress proceedings (see Rückert (1999a)).
The paper is a general presentation and defence of the dialogical approach from the standpoint of logical pluralism. It counters some prejudices against dialogical logic, stresses the advantages of the framework (specifically the distinctions between the game level and the strategic level, between structural rules and particle rules and between logical validity as general validity and logical validity as formal validity), and gives an overview of most of the Rahman/Rückert-papers contained in this collection.6
6 For another, more recent, general presentation and defence of dialogical logic see Rahman/Keiff (2004).
Rahman, S., Rückert, H. and Fischmann, M.: ‘On Dialogues and Ontology. The Dialogical Approach to Free Logic’, Logique et Analyse 160 (1997), p. 357-374
This paper was written together with one of the students who took part in our graduate course
“Erweiterungen der Dialogischen Logik”, Matthias Fischmann. It was first published in the memo series (memo no. 24, October 1998) and then accepted for publication in the international journal Logique et Analyse. In the paper it is shown that reflections on ontological assumptions that lead to so-called free logics can be captured in the dialogical framework by special structural rules that apply to the attack and defence of quantified statements.
Rahman (2001) is an interesting paper that combines ideas about free logics from this paper with dialogical formulations of paraconsistent and intuitionistic logic.
Rahman, S. and Rückert, H.: ‘Dialogische Modallogik (für T, B, S4 und S5)’, Logique et Analyse 167-168 (1999), p. 243-282
This paper provides dialogical formulations of the most basic systems of modal logic. The most important new concept introduced in it is the one of a dialogue context, which corresponds, roughly speaking, to the concept of a possible world used in other approaches.
The paper was first published in the memo series (memo no. 25, November 1998) and was later accepted for publication in Logique et Analyse.
There is some related work which started from this paper: Thus Patrick Blackburn (2001) has generalised the dialogical approach to modal logic to formulate dialogical versions of so-called hybrid logics (modal logic with nominals to name specific possible worlds) and Ulrich Nortmann (2001), by manipulating the rule which prohibits unnecessary moves (the no-delaying-tactics rule), extends the approach to provability logic.. Shahid Rahman (2006) has shown how to deal with so-called non-normal modal logics within the dialogical framework.
Rahman, S. and Rückert, H.: ‘Dialogische Logik und Relevanz’, Universität des Saarlandes, memo no. 27, December 1998
In this paper ideas about relevance are captured by imposing a certain relevance criterion which must be met by the winning strategies for a formula to be relevantly valid. We used here the strictest criterion possible and thus received a logical system with very high standards of relevance. In contrast to that, the best known systems of relevance logic generally use weaker standards of relevance which leads to stronger logics with more valid formulas. For this reason, after publishing the paper in the memo series, we didn’t submit it in this form to an international journal, but had the plan to expand the paper some day in order to add dialogical versions of the standard systems of relevance logic with weaker criteria of relevance, too. This plan has so far not been realised and is a desideratum for future research.
An interesting idea that is not already present in this paper, is the following: Instead of adding criteria of relevance at the strategic level one could build them in already at the game
level itself. This can be done with the help of certain subdialogues which make it possible that a player in the dialogue challenges the other one and asks him to show in this subdialogue that what he stated in the main dialogue meets the criterion of relevance at stake.
Rahman, S. and Rückert, H.: ‘Dialogical Connexive Logic’, in: Rahman, S. / Rückert, H.
(ed.): New Perspectives in Dialogical Logic, Synthese 127 (1/2) (2001), p. 105-139
We first published a German paper on dialogical connexive logic in the memo series (cf.
Rahman/Rückert (1998b)) and then revised it thoroughly for the English version which was published in Synthese. The English version of the paper was presented by Shahid Rahman and me at the New Perspectives in Dialogical Logic conference (Saarbrücken, June 1999) and by myself at the Logic and Games workshop (Amsterdam, November 1999)7 and at the Connexive Logic and Beyond workshop (Siena, June 2000).
The system of connexive logic developed in this paper by the introduction of a new connective, the connexive conditional, seems to be very close to a connexive logic which was formulated approximately at the same time and completely independently from ours by Graham Priest (1999) who used a model-theoretic semantics. It might be promising to examine more closely whether these two systems of connexive logic, formulated within completely different semantic frameworks, are indeed equivalent to each other.8
Rahman, S. and Rückert, H.: ‘Eine neue dialogische Semantik für lineare Logik’, previously unpublished paper (written in 2000/01)
This previously unpublished paper was written in 2000/01 and presents a dialogical semantics for linear logic. We didn’t submit it for publication because the dialogical semantics only captures one variety of linear logic and the equivalence proofs were still missing. So, we planned to expand this important material to a more comprehensive dialogical treatment of linear logic (the result arguably would be of book size) and publish it then. But owing to the geographic academic separation with Shahid Rahman and myself working at different locations collaboration became more difficult and this project is still awaiting realisation.
From a methodological viewpoint the paper is quite stimulating as it sheds some new light on the nature of non-classical logics. According to one viewpoint one receives new non- classical logical systems by changing the set of structural rules against a fixed set of particle rules. This feature is shared by dialogical logic and so-called substructural logics (see for example Restall (2000)). In the context of substructural logics this methodological idea is sometimes called ‘Došen’s Principle’.9
7 An extended abstract for this talk was published in the workshop reader. Cf. Rückert (1999b).
8 Another very interesting paper on connexive logic is Pizzi/Williamson (1997). For an overview on connexive logic see Heinrich Wansing’s online entry in the Stanford Encyclopedia of Philosophy:
http://plato.stanford.edu/entries/logic-connexive/#Bib.
9 This name was suggested by Heinrich Wansing (1994).
On the other hand linear logic is well known for its many different logical particles.
So, one could formulate an alternative methodological principle according to which new non- classical logics can be received by introducing new logical constants. In Rahman/Rückert (2001b) we referred to this idea by naming it after the inventor of linear logic ‘Girard’s Principle’.
Our paper on linear logic shows that Girard’s logical particles can be reconstructed dialogically by the help of certain structural rules that only apply locally, i.e. with respect to certain particles, and not in general. So what is needed are not new particle rules, but only a specific kind of new structural rules.
This result seems to suggest a certain compromise between the two principles mentioned above: Contrary to how it may first appear, Došen’s Principle and Girard’s Principle don’t contradict each other. One can not only generate new logical systems by either changing the structural rules or by introducing new logical particles, but it is possible to proceed according to both devices at the same time, by introducing new particles with the help of a special kind of structural rules. Further future research about these methodological ideas might lead to more clarity and relevant technical results.
Rückert, H.: ‘Logiques Dialogiques ‘Multivalentes’’, Philosophia Scientiae 8 (2), 2004, p.
59-87
This paper was written for a special issue of the journal Philosophia Scientiae dedicated to logic and game theory, edited by Manuel Rebuschi and Tero Tulenheimo.
As there is no place for truth-values in dialogical logic, it was necessary to add a new concept corresponding to the concept of a truth-value in other approaches in order to be able to give a dialogical reconstruction of many-valued logics. This new concept was the one of a mode of assertion. So, with the help of different modes of assertion it became possible to give a dialogical semantics for the standard systems of many-valued logics without using values. A bit paradoxical perhaps, but it worked!10
10 For other work using games in connection with many-valued logics see for example Giles (1974) and Ciabattoni/Fermüller/Metcalfe (2004)
There are many people to whom I am grateful because of their kindness to me, but let me just mention two of them:
1) Shahid Rahman, the “Pope of Dialogical Logic”, who first introduced me to the fascinating world of logic, who later worked with me to produce the papers in this collection, and, most importantly, who is a good friend on whose support I can always count.
2) My mother Monika Rückert, a strong woman who was and still is always there for me.
This volume is dedicated to her.11
Helge Rückert Mannheim (Germany), April 2007
11 I would like to thank Andrew Morton for orthographic, grammatical, and stylistic improvements, and Alexis Georgi for necessary help with various text processing programs.
Abstract:
The aim of this paper is to present the dialogical approach to logic as an interesting alternative to the model-theoretic approach. In the introduction I plead for pluralism concerning logical systems and logical methodology before giving a short outline of Dialogical Logic. Then, I discuss and reject several prejudices against Dialogical Logic. I present three conceptual distinctions that are characteristic of the dialogical approach to logic (namely, formal vs. general truth, level of games vs. level of strategies and particle rules vs. structural rules) and their fruitfulness for logical research is demonstrated at the hand of some examples.
1 Introduction
This paper was originally published in a volume called Essays on Non-Classical Logic. Any logical system that differs from classical two-valued propositional and first order logic can be called non-classical, for example intuitionistic logic, modal logics, relevance logics and such- like. Non-classical logics in this sense are not the main issue of this paper, even though they play an important part in the argumentation.
Instead I consider different approaches to logic or logical methodology. The goal is to defend the dialogical approach, that – at present – seems to be of secondary interest in logical research.1 I will try to show in what aspects the dialogical approach differs from other logical methods and what its advantages are.
A first dichotomy subdivides logical methodology into syntactic and semantic approaches. Whereas the first deals with axiomatic systems, model-theoretic and proof- theoretic (systems of natural deduction, for example) approaches predominate the second category. Dialogical Logic, together with related approaches like Hintikka’s better known GTS (Game Theoretical Semantics)2 constitutes a third alternative on the semantic side of the dichotomy: game-theoretical logical methodology.
The goal of this paper is not to press the superiority of the dialogical approach to all other semantic methodologies, not even to claim that it alone should be used in future logical research. I will not only plead for pluralism concerning the different logical systems, but also for pluralism concerning the different logical methodologies. This means that it is appropriate to use different logical methods to pursue different aims. In my opinion, for many aims
Rückert, H.: ‘Why Dialogical Logic?’, in Wansing, H. (ed.): Essays on Non-Classical Logic (Advances in Logic – Vol. 1), New Jersey, London, Singapore, Hong Kong 2001, p. 165-185
1 There are some signs that the situation is changing and that there is a new start of research in Dialogical Logic (see Rahman/Rückert (2001a)).
2 Hintikka’s (1998) treatment of branching quantifiers in his IF-Logic (Independence Friendly Logic) receives considerable attention from logicians. It would be interesting to examine whether Hintikka’s ideas can be reconstructed in dialogical terms.
Dialogical Logic is a fruitful alternative to the more common model-theoretic or proof- theoretic approaches and it deserves more attention than it gets at present.
In the sequel I defend logical pluralism, and I give a short introduction to the dialogical approach. Furthermore some prejudices that are – among other things – responsible for the fact that Dialogical Logic often is completely unknown or considered exotic are corrected.
Finally, some peculiarities and some resulting advantages of Dialogical Logic are discussed. I would be happy if this paper could contribute a little bit to change the uncaring attitude towards Dialogical Logic.
1.1 Pluralism concerning Logics and Logical Methodology
I plead for logical pluralism, i.e. I think that among the different logical systems developed so far (and still to be developed), including classical logic, as well as the so-called non-classical logics, there is no system that represents the only correct logic. Thus I do not plead for a certain logic as intuitionistic logic, or a version of relevance logic, but for working with a multitude of formal instruments. Which logical system is the most adequate always depends on the context of application and on the purposes for which one wants to use the formal tools.
Thus, different logical systems are formal instruments for different purposes.
The task of the logician should be – among other things – to develop many different interesting logical systems, in which the motivations and underlying ideas are formally worked out. Especially for these purposes I consider the dialogical approach to logic as very suitable, because it often provides the possibility to formally treat interesting ideas in a very simple way. Another advantage is that such formally treated ideas can often be easily combined.
But I do not want to voice the opinion that the dialogical approach provides the best of all available logical methodologies. Even on the methodological level the choice of tools depends on the purposes. Thus – in many contexts – it could be the best choice to use the model-theoretic approach, a Gentzen calculus or another logical method. However, I do think that Dialogical Logic is a fruitful alternative for logical research, especially for developing and combining new logics, and I think that it could be used fruitfully more often than is presently the case.
1.2 Dialogical Logic: a short Outline
Dialogical Logic was suggested at the end of the 1950s by Paul Lorenzen and then worked out by Kuno Lorenz.3 In a dialogue two parties argue about a thesis respecting certain fixed rules.
The defender of the thesis is called Proponent (P), his rival, who attacks the thesis is called Opponent (O). Each dialogue ends after a finite number of moves with one player winning, while the other loses. The rules are divided into structural rules and particle rules. The structural rules determine the general course of a dialogue game, whereas the particle rules
3 Some of the most important early texts about Dialogical Logic are collected in Lorenzen/Lorenz (1978).
show which moves are allowed to attack the moves of the other player or to defend one’s own moves.
Structural Rules
(SR 0) (starting rule):
The initial formula is uttered by P. It provides the topic of the argumentation.
Moves are alternately uttered by P and O. Each move that follows the initial formula is either an attack or a defence.
(SR 1) (no delaying tactics rule):
Both P and O may only make moves that change the situation.4 (SR 2) (formal rule):
P may not introduce atomic formulas, any atomic formula must be stated by O first.
(SR 3) (winning rule):
X wins iff it is Y’s turn but he cannot move (either attack or defend).
(SR 4i) (intuitionistic rule):
In any move, each player may attack a (complex) formula asserted by his partner or he may defend himself against the last attack that has not yet been answered.
or
(SR 4c) (classical rule):
In any move, each player may attack a (complex) formula asserted by his partner or he may defend himself against any attack (including those that have already been defended).5
4 This rule replaces Lorenz’ Angriffsschranken. It still needs to be made clear on a formal basis.
5 Both in SR 4i and SR 4c we have stated the so-called symmetric versions of these rules. It is possible to diminish the rights of O without any changes on the level of strategies in the following way: For P the rules remain unchanged, but O is only allowed either to defend himself against the last move of P or to attack this move. These versions of the rules SR 4i and SR 4c are called asymmetric, because now P has more rights than O.
Particle Rules
∨, ∧, →, ¬,∀,∃ Attack Defence
¬A A ⊗
(No defence, only counterattack possible) A∧B
?L(eft) ---
?R(ight) (The attacker chooses)
A
--- B
A∨B ?
A ---
B
(The defender chooses)
A→B A B
∀x A ?n
(The attacker chooses)
A [n/x]
∃x A ? A [n/x]
(The defender chooses)
The given structural and particle rules define the intuitionistic (with SR 4i) and the classical (with SR 4c) dialogue games. Validity is defined as follows:
Validity (definition):
A formula is valid in a certain dialogical system iff P has a formal winning strategy for this formula. (To have a formal winning strategy means, that for any choice of moves by your opponent you have at least one possible move at your disposition, so that you finally win.)
It can be shown that with these rules and this definition of logical validity the same valid formulas are obtained as in the common approaches to logic. (To get intuitionistic logic use the intuitionistic rule, and to get classical logic use the classical rule.)6
6 Such proofs can be found for example in Barth/Krabbe (1982), in Krabbe (1985), and in Rahman (1993).
Example 1 (with either SR 4i or SR 4c):
O P
(1) (a→b)∧a 0
(3) a→b (5) a (7) b ((a→b)∧a)→b (0) b (8) 1 ?L (2) 1 ?R (4) 3 a (6) P wins Example 2 (with SR 4c): O P (1) ?n 0
(3) ? 2
(5) Pn 4
(3’) ? 2
∀x(Px∨¬Px) (0) Pn∨¬Pn (2)
¬Pn (4)
⊗
Pn (6)
P wins
Remarks concerning the examples:
Here, as in the following, very simple examples are chosen, so that with the help of just one dialogue it is easy to see whether P has a winning strategy for the formula in question or not.
For example 1 it makes no difference whether you use SR 4i or SR 4c, but the formula in example 2 is only valid in classical logic.
Explication of the notation: The moves of O are written down in the O-column, the ones of P in the P-column. The numbers in brackets on the left and right margin represent the order in which the moves were made. Move number (0) is the thesis that is argued about in the dialogue. Attacks are characterised by numbers without brackets that indicate which move of the partner is attacked. Defences are always written down in the same line as the corresponding attacks. Such pairs of attacks and corresponding defences are called rounds.
(As you can see in example 2, when playing according to the classical rule, it is possible for P to defend himself again against an attack that has already been answered before.
To note down this renewed defence, we repeat the corresponding attack, but please keep in mind, that this is not a move in the dialogue, but only a notational convention, which is indicated by an apostrophe as in (3’) of example 2.)
2 Prejudices against Dialogical Logic
Dialogical Logic is generally regarded as exotic and sometimes even completely unknown.
Here, three prejudices that – among other things – have led to the fact that the dialogical approach is in this unfortunate situation will be formulated and then rejected.7
2.1 ‘Dialogical Logic is a Constructivistic Logic’
It is indeed correct that Dialogical Logic was developed in the context of the constructivist program of the so-called school of Erlangen, and that it was used in these surroundings to defend intuitionistic logic. But it is totally wrong to identify Dialogical Logic with intuitionistic or constructivistic logic. Instead, Dialogical Logic represents a general framework8 within logical methodology, comparable to, say, Gentzen’s sequent-calculus formulations, with the help of which classical, intuitionistic, and many other logics can be examined, just by using different dialogue rules. It seems that among the successors to the constructivistic school of Erlangen there is now a tendency to see Dialogical Logic as an inappropriate tool for the foundation (Begründung) of logic (see Hartmann (1998)).
2.2 ‘Dialogical Logic is Limited to Classical and Intuitionistic Logic’
It is right that with some exceptions9 the dialogical approach to logic was for a long time only worked out for classical and intuitionistic logic.10 But in the last few years, together with Shahid Rahman, I have made several proposals to extend it to other logics, too. We have developed dialogical free, paraconsistent, modal, relevance and connexive logics.11 We have shown that results that were reached within a model-theoretic framework, can also be obtained dialogically. But it is even more interesting that, using dialogical methods, it has been possible to treat of certain notions for which there was no perspicuous semantics, examples being certain systems of paraconsistent, relevance and connexive logics.
7 There might be other prejudices against Dialogical Logic. For instance, it should not be concealed that dialogical logicians made some mistakes when working out and presenting their approach. Naturally these mistakes have had a negative influence on the reception of Dialogical Logic (see Felscher (1986)), but mistakes can often be corrected and in a lot of cases this has already been done.
8 ‘Framework’ is here used in a non-technical sense, of course; I don’t wish to propose Dialogical Logic as rival to, say, LF or ALF.
9 Fuhrmann (1985) for example presents a dialogical reconstruction of Anderson/Belnap’s (1975) First Degree Entailment Logic and in Lorenzen (1987) some steps towards dialogical modal logics can be found.
10 Fuhrmann (1985, p. 51) writes: “Die Diskussion um die Dialogische Logik ist bisher ausschließlich aus dem Blickwinkel klassischer bzw. konstruktiver Positionen geführt worden. Relevanzlogiken und parakonsistente Logiken sind nicht angemessen in Betracht gezogen worden.”
11 See Rahman/Rückert/Fischmann (1997), Rahman/Carnielli (2000) and Rahman/Rückert (1998a, 1999 and 2001b).
It is evident that these few papers about non-classical logics formulated within the dialogical framework cannot compete with the immense literature about non-classical logics treated with the help of model theory. But this does not imply that it would also be impossible to reach a comparable amount of interesting results with the dialogical approach,12 if one invested as much time, work and money in such a kind of logical research.
2.3 ‘Dialogical Logic Complicates Things Unnecessarily’
It is correct that there are several distinctions in Dialogical Logic that cannot be found in other approaches to logic. But these distinctions do not complicate simple logical procedures essentially (for example the test whether a formula is valid in a certain logic still is a trivial matter in most cases), but they offer conceptual possibilities for certain purposes.
The impression that Dialogical Logic is unnecessarily complicated probably arises when one is not used to the dialogical approach. But with some training it is easy to see that playing dialogues and checking formulas for winning strategies is as easy as writing down truth-tables or Beth-tableaux. For some time students at Saarbrücken learned the dialogical approach simultaneously with the common model-theoretical approach to logic, and it is remarkable that at least when treating quantifiers most of them had less trouble with the dialogical approach.
3 Advantages of Dialogical Logic
In Dialogical Logic there are several distinctions that are not available in other frameworks, for example in the model-theoretic approach. These distinctions offer interesting possibilities for logical research. In the sequel I present some examples.
3.1 The Distinction between General and Formal Truth
The standard defintion of logical truth or validity reads as follows: “A proposition A is logically true iff it is true under all interpretations.” Thus, validity is usually seen as a generalisation of truth. It is also called ‘formal truth’, because it depends only on the logical form of the proposition in question and not on the interpretation of its constituents:
“The logical truths, then, are those true sentences which involve only logical words essentially.
What this means is that any other words [...] can be varied at will without engendering falsity.”
(Quine (1976, p. 110))
12 One early interesting technical result obtained by using dialogical methods is Lorenz’ (1978) proof of functional completeness of the intuitionistic connectives, that differs considerably from the model-theoretic proof of McCullough (1971). A comparison of Lorenz’ result with the proof-theoretic completeness theorems of Prawitz (1978) and Zucker/Tragesser (1978) might be of interest.
Now, if we turn to Dialogical Logic the concept of logical truth or validity can be split into two conceptually different notions, namely ‘general truth’, corresponding to the common conception, and ‘formal truth’, where ‘formal’ is understood in a strict sense.
Validity as General Truth
In Dialogical Logic we should think of truth as the existence of a winning strategy for P. To formulate validity as ‘general truth’ within the dialogical framework, we have to introduce so- called material dialogues by adding the following structural rule that replaces the formal structural rule:
SR 5 (rule for material dialogues):
Atomic formulas standing for true propositions may be uttered, atomic formulas standing for false propositions must not be uttered.
With the help of the following definition we have reached a dialogical reformulation of the common view of validity:
Validity as general truth (definition):
A formula is valid iff there is a winning strategy for P in every material dialogue about this formula. This means that P must have a winning strategy for this formula for each possibility of assigning truth-values to the atomic formulas.
Here, it makes no difference whether using the intuitionistic or the classical structural rule. In any case the resulting logic will be the classical one.13 Please notice that when playing material dialogues it is always presupposed that all elementary propositions have a definite truth-value, namely ‘true’ or ‘false’.
Validity as Formal Truth
This is different when we turn to the usual definition of validity in Dialogical Logic, namely validity as formal truth:
Validity as formal truth (definition):
A formula is valid iff there is a formal winning strategy for P in a dialogue about this formula. This means that P must have a winning strategy when the formal structural rule is in use.
This way of seeing validity differs conceptually from the common way, because here validity is not just general truth, validity does not just require the existence of winning strategies for P
13 Besides using the classical structural rule or working with material dialogues, there is a third possibility to reconstruct classical logic in the dialogical approach: A formula is valid in classical logic iff P has a formal winning strategy for it in intuitionistic dialogues with additional tertium non datur hypotheses.
for a certain totality of dialogues, but the existence of a winning strategy for P in a certain kind of dialogue game, namely in a formal dialogue for the formula in question.
Also, it is not presupposed that all elementary propositions have a truth-value, are either true or false. Thus, the dialogical concept of formality is stricter than the usual one: The form is not only independent of the truth or falsity of the elementary propositions, but it is even independent of whether the elementary propositions have truth-values.14
The fruitfulness of the distinction between formal and general truth still needs to be worked out in detail, but it should be obvious that it offers an interesting conceptual instrument for logical research.
3.2 The Distinction between the Level of Games and the Level of Strategies
The distinction between the level of dialogue games where questions of meaning and sense are located and the level of strategies where questions about truth and validity are located is not available in other semantic logical approaches.15 Usually the meaning of the connectives is defined by using concepts of correctness such as ‘true’ and ‘false’, for example. In dialogues it is possible that the players make moves that are bad moves from a strategic point of view, and a player may even lose who could have won if he had played in a different way. This acceptance of strategically bad moves on the level of games seems to be unnecessary, but this is not the case. The level of games is important, too, and stands in its own right.
As an example I will try to show that the distinction between the level of games and the level of strategies allows an almost trivial proof of the disjunctive property for dialogical intuitionistic logic. Usually this proof is rather complicated for Gentzen sequent-calculi that allow more than one formula on the right-hand side of a sequent, but when using dialogical methods it is not.
The disjunctive property of intuitionistic logic says that A∨B is valid iff A is valid or B is valid:
╞ A∨B ⇔╞ A or╞ B
The proof from the right to the left is unproblematic. If P has a formal winning strategy for A or for B, it is evident that he has a winning strategy for A∨B: P starts the dialogue by stating the thesis A∨B, and O attacks it with ‘?’. In order to win P then just has to choose the disjunct he has a winning strategy for. If he has a winning strategy for A (or B) he has to choose A (or B respectively) to answer O’s attack.
14 Thus with respect to the elementary propositions it is not required that there is either a winning strategy for P or for O. What is required is that there is a way of arguing about the elementary propositions. This means, they must have sense or meaning, but not necessarily a truth-value. (For the possibility in Dialogical Logic of separating the senses or meanings of propositions from their having truth-values see the next section.)
15 There might be a few other approaches – as, for example, Martin-Löf’s (1984) Intuitionistic Type Theory – which have built in analogous distinctions.
The proof from the left to the right seems much more difficult, but it also becomes almost trivial in Dialogical Logic, because of the distinction between the level of games and the level of strategies. On the level of games intuitionistic logic is characterised by the so- called intuitionistic structural rule:
(SR 4i) (intuitionistic rule):
In any move, each player may attack a (complex) formula asserted by his partner or he may defend himself against the last attack that has not yet been answered.
The crucial point of this rule with regard to the present argumentation is that P is not allowed to defend himself against an attack of O he has already answered, unless O renews his attack.
In our example this means that P is not allowed to defend himself again with B (or A) against the attack ‘?’ on A∨B if he has already defended himself with A (or B respectively).
Now we go to the level of strategies and recall the definition of validity in the dialogical approach: A formula is valid iff P has a formal winning strategy for it. Thus, from the left to the right the metatheorem of the disjunctive property says, that if P has a winning strategy for A∨B he also has a winning strategy for at least one of the two disjuncts. And to have a winning strategy means for P that he is able to win the dialogue no matter how O plays.
If we look at the beginning of a dialogue with the thesis A∨B, it is clear that the first move of O has to be the attack ‘?’ and that P has to reply A (or B) in the second move. Now, the dialogue continues with an argumentation about A (or B respectively) alone, if O does not renew his attack on A∨B. Consequently, if P has a winning strategy for A∨B, he must also have a winning strategy for at least one of the two disjuncts alone, quod erat demonstrandum.
So we see that with the help of Dialogical Logic the proof of the disjunctive property of intuitionistic logic becomes almost trivial. It results directly from combining the intuitionistic structural rule on the level of games with the definition of validity that is formulated on the level of strategies. Here it is irrelevant whether we use the symmetric or the asymmetric structural rule. As strategy tableau-systems (i.e. decision methods for checking whether P has a formal winning strategy or not) for symmetric dialogues correspond to intuitionistic semantic tableau-systems for Gentzen sequent-calculi in which more than one formula on the right-hand side of a sequent is allowed, we have obtained – by using dialogical methods – a very easy proof of the disjunctive property, and, modulo the equivalence-proof concerning tableaux and dialogue-formulations, also for the Gentzen systems mentioned above.16
16 This argumentation is worked out in detail in Rahman/Rückert (1998/99). This paper also includes a discussion of the philosophical background and consequences.
3.3 The Distinction between the Particle Rules and the Structural Rules
In Dialogical Logic the rules are divided into particle rules and structural rules. The particle rules determine for each logical particle how the corresponding formulas can be attacked and defended. The structural rules determine the general course of a dialogue.17
Now it is very interesting to see that you can get to different logical systems by only changing the set of structural rules while retaining the same set of particle rules. For example, dialogical classical and intuitionistic logic differ only in one structural rule, and you can get to a paraconsistent or a free logic by adding certain structural rules. Thus, the differentiation between particle rules and structural rules makes it very simple to generate new logics by combining certain structural rules.
The division into particle and structural rules is an essential feature also of Display Logic.18 In Display Logic the idea that alternate systems are to be obtained by modifying structural rules against the background of a fixed set of connective rules is sometimes referred to as Došen’s Principle:
“[T]he rules for the logical operations are never changed: all changes are made in the structural rules.” (Došen (1988, p. 353))
Thus, the proof-theoretic approach of Display Logic working with modifications of Gentzen style sequent-calculi and Dialogical Logic apply the same methodology for presenting non- classical logics and combinations of them. In this respect both could be seen as parts of a common enterprise. In any case, a detailed comparison of Dialogical Logic and Display Logic remains an interesting task.
Here, I will first present the main ideas of dialogical free, paraconsistent, modal and relevance logics, and then I will discuss the possibilities of combining them.19
Dialogical Free Logic
The main idea of dialogical free logic is to make the quantifiers stronger so that they carry existential import: if a player in a dialogue uses a certain constant to defend an existential quantifier or to attack a general quantifier he simultaneously admits that this constant denotes an existing object. As in formal dialogues P is only allowed to use information that O has admitted first, we can add the following structural rule to our usual set of rules to define free (and inclusive) formal dialogue games:
17 Thus every rule governing a dialogue game that is no particle rule is called a structural rule. Perhaps it is appropriate to refine the category of structural rules, for example the class of formal rules (for atomic formulas, for attacking negations or for introducing new constants) seems to be of particular interest.
18 Display Logic was inaugurated by Belnap (1982). Further work has been done for example by Wansing (1998).
19 Dialogical connexive logic is not appropriate for my present purpose, because there not only the structural rules but also the particle rules, at least for the subjunction, have to be extended (see Rahman/Rückert (2001b)).
SR 6 (formal rule for constants):
Only O may introduce constants. (A constant is introduced iff it is used for the first time to defend an existential quantifier or to attack a general quantifier.)
With the help of this rule we get a basic dialogical system of free logic.20 A simple example shows that in the so defined dialogical free logic fewer formulas are valid than in non-free logics:
Example 3 (free logic):
O P
(1) Pn 0 (3) ? 2
Pn → ∃xPx (0)
∃xPx (2)
O wins
P loses because he is not allowed to defend himself against the attack in move (3): No constant has been introduced so far. (Note that n has not been introduced even if it appears in the dialogue.)
Dialogical Paraconsistent Logic
The main point concerning paraconsistent logics is that not every contradiction should make the whole system trivial. This means ex falso sequitur quodlibet should not be valid in general. In Rahman/Carnielli (2000) several proposals for extending Dialogical Logic are made that lead to this effect. An interesting idea is that only contradictions on the atomic level should be excluded from implying anything.21 For this purpose we have to add the following structural rule:
20 This basic system is similar to some versions of so-called outer domain systems of free logic (see Bencivenga (1986)), but it is inclusive, i.e. it allows the universe of discourse to be empty. It is also possible to invent more sophisticated dialogical free logics, for example with an indefinite number of pairs of quantifiers (see Rahman/Rückert/Fischmann (1997)).
21 One way to defend this approach is to distinguish between two sorts of negations, a negation de re and a negation de dicto (see Sinowjew (1970)). If there is a contradiction with a negation de dicto (possible only on the atomic level), the system does not become trivial, because in dialogical paraconsistent logics such a contradiction gets isolated in a certain sense, so that it cannot be used to infer anything (see Rahman (2001)). Wittgenstein is a precursor of such an idea: “You might get p.~p by means of Frege’s system. If you can draw any conclusion you like from it, then that, as far as I can see, is all the trouble you can get into. And I would say, “Well then, just don’t draw any conclusions from a contradiction.’’’’ (Diamond (1976, p. 220))
SR 7 (formal rule for negative literals):
P is allowed to attack the negation of an atomic formula iff O has attacked the same formula before.
Example 4 (paraconsistent logic):
O P
(1) a∧¬a 0 (3) a
(5) ¬a
(a∧¬a) → b (0)
1 ?L (2)
1 ?R (4)
O wins
P loses because he is neither allowed to defend himself with b (because of SR 2) nor to attack
¬a with a because O has not done the same before.
Dialogical Modal Logic
To get to dialogical modal logics it is not sufficient to change the set of structural rules, but here we have to introduce new particle rules for the modal operators ‘necessary’ and
‘possible’. The new concept of a dialogue context also has to be introduced.22 In dialogical modal logics a dialogue game is played by stating formulas in different dialogue contexts. If a player states ‘necessarily p’ in a dialogue context n he commits himself to state p in all dialogue contexts that are admissible from n. Correspondingly, if a player states ‘possibly p’
he commits himself to defend p in at least one admissible dialogue context.
22 Dialogue contexts are the dialogical counterparts to possible worlds in model-theoretic modal semantics.
Particle Rules for the Modal Operators
, ◊ Attack Defence
A
(in dialogue context α)
?
(in an admissible dialogue context β,
chosen by the attacker)
A
(in β)
◊A
(in dialogue context α)
? (in α)
A
(in an admissible dialogue context β,
chosen by the defender)
For each modal logic system we must determine by supplementary structural rules which dialogue contexts are admissible. In Rahman/Rückert (1999) the dialogical approach to modal logics has been worked out for classical and intuitionistic versions of T, B, S4 and S5. It should not be very difficult to reconstruct other modal logic systems by mere modifications of the structural rules.23
Dialogical Relevance Logic
In Rahman/Rückert (1998a) we have made a proposal for a very strict dialogical relevance logic (even uniform substitution is no longer valid).24 In this case we get to this non-classical logic not by changing the set of rules on the level of dialogue games, but by formulating a criterion that the winning strategy of P has to fulfil: The winning strategy should contain no redundant parts.
23 Nortmann (2001) presents a dialogical version of the modal logic system G.
24 The name ‘dialogical relevance logic’ is perhaps misleading, because our system differs considerably from the usual systems called relevance logic. Our system is rather a formal logical version of the well-known Gricean maxims for conversation (cf. Grice (1989)). It still needs to be examined whether commonly known systems of relevance logic can be formulated in dialogical terms by changing the set of structural rules. One step in this direction was taken by Fuhrmann (1985).
Relevant validity (definition):
A formula is relevantly valid iff there is a formal winning strategy for P that contains no redundant parts (this means that P would have no winning strategy left if he didn’t have all of his possible move choices at his disposition).
Example 5 (relevance logic):
O P
(1) a∧b 0 (3) b
(a∧b) → b (0)
b (4)
1 ?R (2)
P wins (but not relevantly)
Even if P has a formal winning strategy for this formula, it is not relevantly valid, because he does not need to use the attack ‘?L’ against move (1) of O to win.
Combinations of Logics
In dialogical free and paraconsistent logic only the set of structural rules was changed, in dialogical modal logic new particle rules and new structural rules had to be added and in relevance logic a new definition of relevant validity was introduced. But it is decisive for the possibility of combining different non-classical dialogical logics, that the changes are all at different locations in the dialogical framework, so that it is no problem at all to combine them.
The ideas presented here already offer two to the power of five possibilities for different dialogical systems (please keep in mind that you can always choose between the intuitionistic and classical structural rule), for example intuitionistic non-free modal relevance logic or classical free non-modal relevance logic.
It is clear that not all of these different possibilities are equally interesting. Perhaps some of them do not have a field of application at all. But at least some of them seem to be very interesting: in Rahman (1999a) a combination of free and paraconsistent logics is proposed to treat fictional entities. This idea is worked out in more detail in Rahman (2001), where a further combination with intuitionistic and classical features is developed: The players can argue classically with regard to existing objects but have to use the intuitionistic structural rule concerning fictions.
4 Concluding Remarks
With this paper I intended to defend the dialogical approach to logic in a general way rather than work out one dialogical logic system extensively. Thus a more detailed examination of every example and step in the argumentation was not possible. If the reader has the impression that many themes have not been treated in detail, he should have a look at the corresponding articles referred to in the text. In this paper it was my goal to correct some prejudices
concerning Dialogical Logic and to stimulate more interest in it by presenting some possibilities for logical research that the dialogical approach offers.25
25 I wish to thank Shahid Rahman (Lille), Goeran Sundholm (Leiden) and an anonymous referee for helpful suggestions, and Karl Hudson for correcting my English.
Shahid Rahman, Helge Rückert and Matthias Fischmann
Abstract:
Being a pragmatic and not a referential approach to semantics, dialogical logic does not understand semantics as mapping names, propositions and relationships into the real world to obtain an abstract counterpart of it, but as dealing (handeln) with them in a particular way. This allows a very simple formulation of free logic the core of which can be expressed in a nutshell, namely: in an argumentation, it sometimes makes sense to restrict the introduction of singular terms in the context of quantification to a formal use of them. That is, the proponent is allowed to use a constant iff this constant has been explicitly conceded by the opponent.
More technically, we show a new, dialogical way to build free logic systems for first-order logic with classical and intuitionistic features and present their corresponding tableaux.
1 Introduction
1.1 Free Logics
The proposition “God does not exist” contains a paradox sometimes referred to as Plato’s beard: if God does not exist and the proposition should be true, standard referential semantics for quantified logic fails to give meaning to the name “God”. But, given compositionality, since the meaning of a sentence is combined from the meanings of its parts, “God does not exist” does not evaluate.
It is easy to see that related difficulties appear in every formula that contains singular terms. In standard logic, it is impossible to state that God is either good or evil without presupposing his existence, or that the round square is round, or even that the flying horse can fly. We always get caught by the lack of (referential) meaning of some of the parts of the sentences.
Several modifications to standard logic have been proposed to deal with formulae containing referential gaps while still defending reference (for instance, Russell’s (1905) theory of descriptions renders them false by translating “The P is Q” to
∨
(x) (P(x) ∧∧
(y)(P(y) → x=y) ∧ Q(x)). But they all suffer the same shortcoming: Logic and ontology are conceptually distinct topics, but reference as a tool for giving meaning to formulae necessarily mixes them up. Only such names can be used to build meaningful formulae that denote some entity.1
Rahman, S., Rückert, H. and Fischmann, M.: ‘On Dialogues and Ontology. The Dialogical Approach to Free Logic’, Logique et Analyse 160 (1997), p. 357-374
* My work on this article has been supported by the Fritz-Thyssen-Stiftung which I wish to thank expressly.
1 Cf. Hintikka (1958). Castaneda (1974) wrote about Plato-Meinongian objects. Though he was mainly interested in ontology and not in logics, the formal nature of his style would be useful to develop non-free logical systems to argue about fictive or contradictory entities.
An interesting class of more or less non-standard approaches to give meaning to singular terms is called free logics. The name is due to Lambert (1962 and 1963) who first used it in 1960 for a system free of any existence assumptions about entities.2 Bencivenga (1986) gave one of the most fruitful definitions in his introduction to free logics:
“A free logic is a formal system of quantification theory [...] which allows for some singular terms in some circumstances to be thought of as denoting no existing object, and in which quantifiers are invariably thought of as having existential import.”
Interestingly Quine’s (1980) ontological criterion that “to be is to be the value of a bound variable” still holds; it even becomes the fundamental principle of most free logics. Existence is still part of our logic, expressed by the quantifiers: “everything” means “everything that exists”, “anything” is “any existing thing”. The crucial difference is that existence does not influence the truth or falsity of sentences containing singular terms that escape the scope of quantification in the same way as in Quine’s point of view.
Many different free semantics have been presented, but basically they can be split into two main categories:
1. In outer domain systems, reference still holds, but “God” may either denote an object from a domain of existents DE or from a newly created domain of non- existents DN. In some flavours of outer domain free logics, DN is more complex and can itself contain more than one domain.
2. Systems based on modality are somewhat more sophisticated. Every singular term either denotes an existing entity or it does not denote at all; there simply are no such things like non-existents. To make a non-denoting singular term meaningful, models related to Kripke’s possible worlds are used in one or the other way (e.g.
by mapping formulae containing referential gaps to truth values using a convention like van Fraassen (1966a) did, or by directly evaluating the referential gaps in possible worlds like in Bencivenga (1981)).
We will not go into the details of free logics based on reference nor will we discuss advantages and drawbacks (cf. Read (1995, chapter 5)). Instead, we will present a different approach that proposes a pragmatic view on the relations between ontology and logic. We base this approach on dialogues.
1.2 Dialogues
Dialogical logic, suggested by Paul Lorenzen in 1958 and developed by Kuno Lorenz in several papers from 1961 onwards (cf. Lorenzen/Lorenz (1978)3), was introduced as a
2 Although free semantics had not yet been given birth to then and Lambert’s concerns were purely syntactic this obviously was what he had in mind when designing his proof system. Cf. also Leonard (1956).
3 Further work has been done by Rahman (1993) in his PhD thesis.
